Counting Figures in Planar Random Configurations

Random configurations are considered that are generated by a Poisson process of figures in the plane, and a recent result is used to derive formulae for the estimation of the number of figures, and their mean area and perimeter. The formulae require merely the determination of the area, the perimeter, and the Euler-Poincaré characteristic of the random configurations in a fixed field of view. There are no similar formulae for the standard deviations of the estimates; their magnitudes in typical cases are therefore assessed by Monte Carlo simulations

The variance of a poisson process of domains

A familiar relation links the densities that result for the intersection of a convex body and straight lines under uniform isotropic randomness with those that result under weighted randomness. An extension of this relation to the intersection of more general domains is utilized to obtain the variance of the n-dimensional measure of the intersection of two bodies under uniform isotropic randomness. The formula for the variance contains the point-pair distance distributions for the two domains — or the closely related geometric reduction factors. The result is applied to derive the variance of the intersection of a Boolean scheme, i.e. a stationary, isotropic Poisson process of domains, with a fixed sampling region

Risk projections in time

The nominal risk coefficients for radiation induced cancer are largely based on the follow-up of the mortality from solid cancers among the atomic bomb survivors. For those who have been exposed as adults, the observations are essentially complete, and the risk estimates are, therefore, firmly based on observations. Those who have been exposed as children, have still not reached the age of high cancer incidence. Their observation is, therefore, still incomplete, and the risk estimates are correspondingly uncertain. The modelling of risk has predominantly been based on the postulate, that the relative risk (i.e. the actual cancer rate divided by the age specific normal rate) depend on dose and on age at exposure, and that it does not decline with time since exposure. The high relative risks observed at young ages lead, therefore, with this type of model, to high estimates of life time attributable risk. The ICRP recommendations contain these high risk estimates for young ages at exposure; the high sensitivity of children and juveniles has, indeed, become one of the basic tenets of radiation protection. It is here shown that these conclusions are still hypothetical, because they are merely a matter of the choice of the model. An alternative model assumes a dependence of the excess relative risk on age attained, rather than age at exposure. This model fits the data equally well, and predicts no increased risk for young ages at exposure. A decision between the two models is not possible at present, it will have to await the continued follow- up of those who survived the atomic bombs as children. The ICRP has been criticised for postulating a dose reduction factor (DDREF) in their nominal risk coefficients. If they abandoned this factor, and used the age attained model, rather than their present model, their numerical risk coefficients would remain unchanged